We develop a reduced-order theoretical framework for inertial spheroidal particles in turbulent Navier-Stokes flow, representing their orientation with a single unit vector (the director) instead of using a full rotation matrix. This symmetry-based simplification, analogous to Jeffery's classical description of neutrally buoyant spheroids, yields a compact set of evolution equations with fewer degrees of freedom. In the limit of small inertia, the director obeys Jeffery's equation which we consider as the equation on direction of distance between two infinitesimally close trajectories of a chaotic system. This implies that the particle's director is uniquely determined by its position, defining a director field that, in turn, induces a smooth, weakly compressible effective flow through which the spheroids move. This effective flow is distinct from the underlying incompressible turbulent flow, thereby generalizing Maxey's result for spheres. In homogeneous, anisotropic turbulence with zero mean fluid flow, the effective particle flow can acquire a nonzero mean; for example, in rotating turbulence, particles drift uniformly along the rotation axis. We show analytically that prolate spheroids align with the major Lagrangian stretching direction of the turbulent flow, whereas oblate spheroids align with the major Lagrangian compression direction. In spatially inhomogeneous flows (e.g., a wall-bounded channel flow), particle anisotropy can alter turbophoretic drift, possibly reversing the usual migration toward quiescent regions. We also show that inertial spheroids exhibit preferential concentration. Finally, when gravity is included, we derive analytic expressions for the mean settling velocity in the weak-gravity regime. This settling velocity increases monotonically with the particle aspect ratio, from $8/9$ of the Stokes settling velocity of an equivalent sphere for disklike particles to $4/3$ for rodlike particles. In the strong-gravity limit, the driving of spheroids by the Navier-Stokes turbulence can be effectively described as white noise. We show that very small oblate spheroids settle edge-first, but small ones settle broadside-on. The orientation-dependent settling velocity field is spatially varying and weakly compressible, providing a new mechanism for multifractal preferential concentration. Thus, this director-based framework provides a unified, physically transparent generalization of Jeffery's and Maxey's theories, yielding multiple new analytical predictions for the transport of nonspherical particles in isotropic, anisotropic, and stratified turbulent flows and environments.

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• Fluid-particle interactions
• Particle-laden flows